Winter Term, 2016
Name:
Section: B
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 215 Set Theory
Midterm 1 February 10, 2016
SOLUTIONS
Instructor:
Dr. P. Zhang
Time Allowed: 50 minutes
Total Value: 100 marks marks
Number of Pages: 4 plus cover p
MA215: WINTER 2014
SOLUTIONS FOR ASSIGNMENT 6
(1) Let n be a positive integer. Define f : Bn cfw_0, 1n by
f (A) = (a1 , a2 , an ),
where
(
ai =
if i A
otherwise.
1
0
(a) Use this function to prove that Bn  = 2n .
Solution: Define a function g : cfw_0, 1
MA215: WINTER 2014
SOLUTIONS FOR ASSIGNMENT 4
(1) Let Z = Z \ cfw_0. Define the relation on Z Z by the rule (p, q) (r, s) if and
only if ps = qr.
(a) Prove that is an equivalence relation on Z Z .
Solution: To prove this is an equivalence relation, we che
MA215: WINTER 2014
SOLUTIONS FOR ASSIGNMENT 3
(1) For positive integers n and k, define
Bn,k := cfw_A P(cfw_1, 2, . . . , n) : A = k.
Define the relation on Bn,k by the rule A B if and only if A B = . We say
that the elements A, B, C P(cfw_1, 2, . . . ,
MA215: WINTER 2014
SOLUTIONS FOR ASSIGNMENT 1
(1) Consider the following sets:
A = cfw_n N : n = m2 for some m N,
B = cfw_n N : n = m2 for some even m N,
C = cfw_n N : n is divisible by 4.
Determine, with proof, whether or not each of the following statem
MA215 Test 1, Fall Term 2007
Page 1
Disclaimer: Tests and exams from previous oerings of mathematics courses are posted to the online Exam
Bank, administered by the Mathematics Assistance Centre, as a courtesy to WLU students. Past exams
should be used f
MA215 Test 2, Fall Term 2007
Page 1
Disclaimer: Tests and exams from previous oerings of mathematics courses are posted to the online Exam
Bank, administered by the Mathematics Assistance Centre, as a courtesy to WLU students. Past exams
should be used f
MA251 Set Theory (2016W)
Lecture on September 29, 2016
Construction of the set of intergers from the set N
In the following thirteen parts of the question, we will prove the existence of
integers and verify their essential properties. (Recall that we prov
'IIlenrem 6.10 (SchmederBematein: If A i B and B1 11 lAl, then A = B.
Theorem 6.12 (Latinr of Trlcllutonly): For any two sets A and B, enaetljyr one of the following is true:
IAI <1 IBI, A= IBI, IAIHBI
By Centers theorem, R0 < 2 and, as noted
Theorem 6.14: The addition and multiplication of cardinal numbers satisfy the properties
Cardinal numbers Sets
(a+m+7:a+0$w (n Muenuleuqu)
a+B=+a (a AUB=BUA
(a)7=a(,67) (3) (AXB)><CHA><(B>< C)
a=x (a AmeBxA
oz(;6+'y)=a+ory (5) Ax(BUC)=(A><B)U(AxC)
Ifog,th
EQUIVALENCE RELATIONS
Denition. A relation R on a set S is called an equivalence relation if it
is reexive, symmetric and transitive; that is,
cfw_I For every n E S, oils.
cfw_2 If Rb, then bRa.
(3] IfRb and bile, then aRc.
PARTITIONS
Let S be a nonempty
4.6 RECURSIVELY DEFINED FUNCTIONS
A function is said to be recursiveiy dened if the function denition refers to itself.
(1) There must be certain arguments, called base values, for which the function does not
refer to itself.
(2) Each time the function do
I RDERING 0F CARDINAL NUMBERS
Denition 6.6: Let A and B be sets. We say that MI E IBI
if A has the same cardinality as a subset of B or, equivalently, if there exists a oneto
one (injective) function f: A :~ B.
As expected, Mi 5, B is read: The cardin
LEMMA A countable union of nite sets is countable.
Theorem 6.5: Let A1,A2,A3, . . be a sequence of pairwise disjoint denumerable sets. Then the union
A1 UAZ U A3 U  .  = U(A, : :76 P is denumerable.
Corollary 6.6: A countable union of countable sets is
MA215: WINTER 2014
SOLUTIONS FOR ASSIGNMENT 2
(1) Let p and q be polynomials in two variables. Define the zero set of p to be
Zp := cfw_(x, y) R2 : p(x, y) = 0.
The following questions deal with zero sets.
(a) Formulate and prove a conjecture about the re
MA215: WINTER 2014
SOLUTIONS FOR ASSIGNMENT 5
(1) In the following problems, is the relation on Z Z defined by (p, q) (r, s) if
and only if ps = qr. Recall that Q = Z Z / . We defined multiplication of two
elements of Q by
[(p, q)][(r, s)] = [(pr, qs)],
a
Proposition.
A.
(5) A = B if and only if
Definition. We call the set of all subsets of a set A the power set of A,
Definition. The number of elements in a set A is denoted by A.
Cantors Theorem. For any set A, A < 
. (to be proved!)
Question: Is the
Principle of Mathematical Induction I: Let A(n) be an assertion deﬁned on P, that is, A(n) is true or
false for each integer n 2 1. Soppose A(n) has the following two properties:
(1) AG) is true.
(2) A(n + l) is true whenever A(n) is true.
Then A(n) is tr
Axiomatlc Development or Set Theory
Any axiomatic development of a branch of mathematics begins with:
(1) undefined terms (in set theory, “element” and “set”),
(2) undefined relations (in set theory, “element belongs to a set”),
(3) axioms relating the un
Lecture 1
Definition 1 A set is a welldefined collection of objects; the
objects are called the elements of the set.
Examples of sets:
(a) Students of MA215 this semester
(b) All odd positive integers, i.e., 1, 3, 5, .
(c) The solutions of the equation 2
MA251 Set Theory (2016W)
Assigned Problems (No. 2)
SOLUTIONS
1. A relation R is defined on Z, the set of all integers, by xRy if 11x 5y is an even
integer. Prove that R is an equivalence relation on Z. Find the partition determined by
R.
Proof. 1) (reflex
Winter Term, 2015
Name:
Section: A
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 215 Set Theory
Midterm 1 February 10, 2015
Instructor:
Dr. P. Zhang
Time Allowed: 80 minutes
Total Value: 100 marks marks
Number of Pages: 4 plus cover page
Instru
Fall Term, 2005
olukions
Name:
Student Number: 
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 215
Test I

Set Theory
 October
5, 2005
Instructor: Dr. A. Bonato
Time Allowed: 80 minutes
Total Value: 40 marks
Number of Pages: 5 plus cover page
MA251 Set Theory (2016W)
Assigned Problems (No. 1)
SOLUTIONS
1. The following story was told by a Chinese philosopher Han Fei Zi (around 281 233 BC).
An armorer of the State of Chu boldly claimed to make the best spears
and shields.
"My shields are so str
Correction of errors/typos in the text
Chapter 1
p.16: Q.1.7 (c)
A \ C = f5g
p.17: Q.1.12
Prove: BnA = B \ Ac .
p.20: Q.1.21(c)
n(C) = 0:
p.31: Q.1.74
Prove: 1 + 3 + 7 +
+ (3n
2) =
n
(3n
2
1).
Chapter 3
p.69: Example 3.5
C = fx; y; xg
p.71: Example 3.7(b)
MA251 Set Theory (2016W)
Assigned Problems (No. 1)
1. The following story was told by a Chinese philosopher Han Fei Zi (around 281 233 BC).
An armorer of the State of Chu boldly claimed to make the best spears
and shields.
"My shields are so strong; they
MA251 Set Theory (2016W)
Assigned Problems (No. 2)
1. A relation R is defined on Z, the set of all integers, by xRy if 11x 5y is an even
integer. Prove that R is an equivalence relation on Z. Find the partition determined by
R.
2. A relation R is defined